Uniform Continuity of Continuous Functions on Compact Metric Spaces

Daniel Daners
Preprint (PDF) June 2014
American Mathematical Monthly 122 No 6 (2015), 592
Original article available at doi:10.4169/amer.math.monthly.122.6.592


(This is intended as a classroom note) We give a direct proof of the fact that a continuous function on a compact metric space is automatically uniformly continuous. The proof is based on standard theorems from metric spaces: The inverse image of a closed set under a continuous function is continuous, the product of two compact metric spaces is compact, and every real valued continuous function attains a minimum on a compact set.

A preprint (PDF) is available or you can go to the original article.

AMS Subject Classification (2000): 54C05, 26A15